Questions tagged [computational-geometry]
Using computers to solve geometric problems. Questions with this tag should typically have at least one other tag indicating what sort of geometry is involved, such as ag.algebraic-geometry or mg.metric-geometry.
503 questions
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Upper bounds on art gallery problems using independent witnesses
Given a polygon $P$, the art gallery problem looks to find a smallest set of points that sees all of $P$. One way of bounding the number of guards necessary (from below) is to find a largest set of ...
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Intersection of Polyhedra
I'm writing a collision detection algorithm, and so far I've been using Joseph O'Rourke's book "Computational Geometry in C" as reference. It outlines an algorithm to determine whether a point is ...
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Area-preserving map between rectangles and fat polygons
Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle ...
- 1,125
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How to find the minimum number of hyperplanes to define a convex hull?
I have the following problem:
I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq ...
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Questions on Discrete Exterior Calculus in numerical computing
I have several questions about Discrete Exterior Calculus (DEC) in numerical methods for solving partial differential equations in physics:
(Discrete Exterious Calculus is a newly developed subject ...
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Convex hull in a discrete space [closed]
I know some algorithms which compute the convex hull in a continuous space.
Are there efficient algorithms to compute it in a discrete domain?
For example in 3D discrete space, given the blue points, ...
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largest diameter of intersection of two balls
Two closed balls with a common radius are positioned so that the centre of either ball is on the boundary of the other.
I am interested in the extremal diameter of their intersection, in an arbitrary ...
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How many dihedral angles need to be specified to uniquely specify a triangulated polyhedron?
Suppose you are given a simplicial complex $K$ homeomorphic to the sphere and for each each edge of the complex a label specifying a length of that edge (this gives us a polyhedral metric on $K$). In ...
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Algorithm for reporting all triangles with unique interior point
What is known about the complexity of and/or practical algorithms for reporting all triplets of points from finite set of at least four points of which no three are collinear in the Euclidean plane, ...
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Computable link invariants
I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...
- 647
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Area ratio of a minimum bounding rectangle of a convex polygon
Take a convex polygon $P$ in the plane, and find its minimum area bounding rectangle, $R$. I'm interested in the ratio of the area of $R$ to the area of $P$. The ratio has a minimum of $1$ for ...
- 1,182
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Fitting one Polygon in another
I have two Polygons A and B and I want to find the position, rotation and scale of B, so it fits into A and has the maximum Area possible. Also both can be concave.
I did some research but couldn't ...
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Volume of a finite union of overlapping balls?
Suppose I have finite list of $n$ 3-dimensional balls, specifying their positions and radii. The balls can have non-empty intersections.
Is there an algorithm to compute the volume of the region ...
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Algorithm for Reconstructing Point Sites from a Voronoi Diagram
how can one construct a finite set of points in the euclidean plane from its Voronoi Diagram and, what is the complexity of the problem?
- 13.2k
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Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?
Is there an algorithm to construct a polyhedron containing all points in space for which there exists no ray to infinite not intersecting a given polyhedron?
In 2D, we could consider polygons. For ...
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Largest inscribed rectangle inside a convex polygon
It has been proved by Radziszewski in this paper
K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. ...
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intersection of convex and non-convex polyhedra
I am trying to find the best appropriate way to intersect polyhedra which may be non-convex.
The number of vertices that build the polyhedron is hence always small (up to 20 or so).
The ...
- 143
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convex polyhedron in the unit cube
Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where $\...
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Decomposing a polygon with holes
It is known that given a polygon $P$ with holes it is NP-hard to find a decomposition of $P$ into convex polygons, s.t. their number is minimized (even if Steiner points are allowed).
I am wondering ...
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Numerical solution for a system of multivariate polynomial equations
Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q_1, q_2, q_3, q_4$ (i.e. polynomials with all the terms such as $q_1^6, q_2^6, q_2^4 q_3^2$):
$P_k(q_1, q_2, q_3, q_4) = 0$ ...
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335
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Software for finding conjugates in the braid group
The conjugacy problem for the braid group was solved by Garside, and gives an algorithm for determining whether two braids are conjugate. Since this algorithm is rather tedious, I was wondering if ...
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Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
- 1,677
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A brief question about the "Eight Queens" Puzzle
The classical Eight Queens puzzle asks whether it is possible to arrange $ 8 $ queens on an $ 8 \times 8 $ chess board, so that no two queens attack each other.
It is well-known that such ...
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Breaking a rectangle into smaller rectangles with small diagonals
Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
- 4,049
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Polyline averaging
I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...
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Computational geometry, tetrahedron signed volume
Short version: I'm trying to compute the orientation of a triangle on a plane, formed by the intersection of 3 edges, without explicitly computing the intersection points.
Long version: I need to ...
- 143
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145
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Complexity of Determining the Edges of Planar Convex Hulls
Question:
can the set of edges that resemble the convex hull ($CH$ for short) of $n$ points in the euclidean plane be determined in $O(n)$ time?
I know that the time complexity of determining the $CH$...
- 13.2k
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Complexity of Random Delaunay Triangulation in 3D
My question:
Is the number of cells in a three-dimensional Poisson-Delaunay triangulation with $n$ vertices $\mathcal O(n)$ with probability one?
which is equivalent to the question
Is the ...
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What properties does generalized Delaunay triangulation have?
Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
- 18.9k
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Finding the "top" or "bottom" vertex of a simplex
A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v_1 \ge p_1$, ... , $v_n \ge p_n$). Similarly, $v$ is a "bottom" ...
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How can I efficiently determine which side of a line segment is internal to the polygon?
As part of a larger analysis I have a need to break a polygon into it's individual line segments and mark which side is "inside" of the polygon. If your curious this is going to be fed into a big ...
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Left and right halves of convex curve
Let $S$ be a set of $n$ points in the plane in general position (no 3 on a line), $n$ even.
A halving line is a line through $2$ points of $S$ that partitions $S$ into 2 equal parts ($(n-2)/2$ points ...
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Algorithm for grouping tetrahedra from Voronoi diagram
I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
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Implementation of Koebe–Andreev–Thurston circle packing?
The circle packing theorem (Koebe–Andreev–Thurston theorem) claims for a planar graph, we can pack disjoint circles, such that: the circles correspond to vertices and the disks are tangent if the ...
- 1,465
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Hyperrectangle partition of set of overlapping hyperrectangles
I have a set of $n$, $d$-dimensional hyperrectangles which may be overlapping in arbitrary ways. I would like to partition the area covered by this set into a set of non-overlapping hyperrectangles. ...
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Finding integer points on an N-d convex hull
Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is ...
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$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$
I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
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Reference: Packing under translation is in NP
I am looking for a reference for a result that I am aware of.
Let me describe the result.
Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP
time, if $p_1,\ldots,p_n$ can be ...
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Points on lines with prescribed distances to each other
Given three lines $l_a, l_b, l_c$ in $\mathbb {R}^3$ and three positive numbers $a, b, c>0$ I would like to find points $A, B, C$ on $l_a, l_b, l_c$ respectively, such that the side lengths of ...
- 2,286
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Minimum number of rectangles in a polygon
Given a polygon and dimension $d$, find a minimum partition of rectangles that has either of its dimensions equal to $d$.
Example:
Consider the following diagram:
I want to cover maximum shaded ...
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How to compute the tangent space of a quotient by a finite group
Let $I\subseteq R:=\mathbb C[x_0,\ldots,x_n]$ be a homogeneous ideal defining a subscheme $X\subseteq\Bbb P^n$. As in my previous question, the permutation group $\mathfrak S_{n+1}$ acts on $R$ by ...
- 4,902
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Area of a Convex Polygon (Described via Half-Planes)
The intersection of a collection of half-planes describes a convex polygon, whose vertices can be constructed in $O(n \log n)$ time using a divide-and-conquer approach (e.g., intersect each half-plane ...
- 3,059
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414
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Computing places over x in F/K(x)
Let $F$ be a function field of "transcendental degree one" over its full constant field $K$. Let $x \in F \backslash K$. We know the divisor of $(x) = (x) - (1/x)$ in $K(x)$. Could you please give me ...
- 601
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Checking if a binary vector lies in the affine span of given binary vectors
Let $x_1,\ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors in ${\mathbb R}^D$, assumed affinely independent. Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ is in ...
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Algorithms for covering a rectilinear polygon using the same multiple rectangles
Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...
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Homogeneous system of polynomial equations
Hi all,
Previously I asked a question that currently has no satisfactory answer Least sum squares given constraints on subcomponents
It comes from an engineering problem. I was thinking to formulate ...
- 101
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123
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Cylindrical Decomposition vs Morse decomposition
Suppose I have a polynomial Morse function $f: \mathbb{R}^n \to \mathbb{R}$. Consider the ideal $I(\nabla f)$ generated by the partial derivatives $\partial_i f$, and assume that the real zero-set of ...
- 505
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458
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Generating random polygons from a given triangulation of points
Given a triangulation $T$ of a planar set point $S$, we would like to randomly generate a polygon (hamiltonian cycle) $P$.
However, it has been proved that Hamiltonian Circuit Problem on maximal ...
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Is there a simple criterion to determine if two parallelograms intersect?
Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.
- 979
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split polygon into minimum amount of rectangles and triangles
Hi
is there an algorithm which cuts a polygon into a minimum amount of preferably rectangles and where not possible (e.g. edges) into triangles?
